# On the distributions of scan statistics of a two-dimensional Poisson process

## Sven Erick Alm

### Abstract

Given a two-dimensional Poisson process, X, with intensity $\lambda$, we are interested in the largest number of points, L, contained in a translate of a fixed scanning set, C, restricted to lie inside a rectangular area.
The distribution of L is accurately approximated for rectangular scanning sets, using a technique that can be extended to higher dimensions.
Reasonable approximations for non-rectangular scanning sets are also obtained using a simple correction of the rectangular result.

#### Key words:

Poisson process, scan statistic, approximation
Research supported by the Axel and Margaret Ax:son Johnson Foundation.
1998-04-06, Sven Erick Alm, sea@math.uu.se